Avitabile D, Desroches M, Knobloch E, Krupa M
School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, UK.
MathNeuro Team, Inria Sophia Antipolis Méditerranée, 2004 Route des Lucioles, BP93, 06902 Valbonne cedex, France.
Proc Math Phys Eng Sci. 2017 Nov;473(2207):20170018. doi: 10.1098/rspa.2017.0018. Epub 2017 Nov 8.
A subcritical pattern-forming system with nonlinear advection in a bounded domain is recast as a slow-fast system in space and studied using a combination of geometric singular perturbation theory and numerical continuation. Two types of solutions describing the possible location of stationary fronts are identified, whose origin is traced to the onset of convective and absolute instability when the system is unbounded. The former are present only for non-zero upstream boundary conditions and provide a quantitative understanding of noise-sustained structures in systems of this type. The latter correspond to the onset of a global mode and are present even with zero upstream boundary conditions. The role of canard trajectories in the nonlinear transition between these states is clarified and the stability properties of the resulting spatial structures are determined. Front location in the convective regime is highly sensitive to the upstream boundary condition, and its dependence on this boundary condition is studied using a combination of numerical continuation and Monte Carlo simulations of the partial differential equation. Statistical properties of the system subjected to random or stochastic boundary conditions at the inlet are interpreted using the deterministic slow-fast spatial dynamical system.
一个在有界域中具有非线性平流的亚临界模式形成系统被重新表述为一个空间上的快慢系统，并使用几何奇异摄动理论和数值延拓相结合的方法进行研究。确定了描述静止前沿可能位置的两种类型的解，其起源可追溯到系统无界时对流和绝对不稳定性的开始。前者仅在非零上游边界条件下出现，并对这类系统中噪声维持的结构提供了定量理解。后者对应于全局模式的开始，即使在上游边界条件为零时也会出现。阐明了鸭轨道在这些状态之间非线性转变中的作用，并确定了所得空间结构的稳定性特性。对流区域中的前沿位置对上游边界条件高度敏感，并使用偏微分方程的数值延拓和蒙特卡罗模拟相结合的方法研究了其对该边界条件的依赖性。使用确定性快慢空间动力系统解释了在入口处受到随机或随机边界条件的系统的统计特性。
